Atmospheric spectral-feature amplitude

An exoplanet transmission spectrum measures a wavelength-dependent transit depth. In a simple opaque-disk geometry with uniform stellar brightness, an effective planetary radius $R_p$ gives $D=(R_p/R_{\ast }{)}^2$, where $R_{\ast }$ is the stellar radius. If stronger atmospheric absorption increases that radius by $\Delta R=N_{H}H$, define $H$ as the atmospheric scale height and $N_H$ as the number of scale heights separating the two spectral levels. The additional occulted annulus givesThe last approximation requires $N_{H}H\ll R_p$. For instance, $R_p/R_{\ast }=0.1$ and $N_{H}H/R_p=0.05$ give exact $\Delta D=0.001025$ and leading estimate $0.001$; these are stipulated geometric ratios.

Since $H=k_{B}T/(\mu m_{H}g)$ in the thin isothermal atmosphere model, larger temperature, smaller mean molecular weight or weaker gravity can increase the signal. The factor $N_H$ is set by the atmospheric opacity, composition and cloud structure; it is not a universal constant. Limb darkening, stellar heterogeneity and instrumental effects also affect the observed transit depth.^[ref]

For a thin hydrostatic isothermal atmosphere, $\Delta D\simeq 2R_p{N}_{H}H/R_{\star }^2$ and $H=k_{B}T/(\mu m_{u}g)$. Hence $T\simeq \mu m_{u}g\Delta D{R}_{\star }^2/(2k_B{R}_p{N}_H)$. A measured exoplanet transmission spectrum amplitude constrains $N_{H}H$, so gravity, mean molecular mass and the number $N_H$ of sampled atmospheric scale heights are additional inputs. In a clear fixed-composition model, $N_H\simeq \log ({\kappa }_{\mathrm{band}}/{\kappa }_{\mathrm{cont}})$; clouds or abundance gradients can change it.